Today’s Uncertainty Wednesday is a quick wrap up to the formalization of beliefs as probability distributions. Let’s first start with a simple question. What probability distribution reflects the least knowledge about a coin? How do you express “I don’t know, the coin could be anything from only heads to only tails to anything in between”? The answer is a uniform distribution as shown here:

The horizontal axis is the probability of the coin coming up heads – ranging from 0 to 1 and the vertical axis is what exactly? Well this depends on whether you paid attention to last week’s explanation of probability density functions (pdf) versus cumulative distribution functions (cdf). Which one is this?

Well it is a pdf, to be precise it is the pdf a uniform distribution over the interval [0, 1], which means that the vertical axis helps us measure the probability over any (sub)interval of the horizontal axis. The area under the curve is the probability. We can easily verify that the total is 1 because 1 x 1 = 1. Now consider the probability between say 0.4 and 0.6. That’s (0.6 - 0.4) x 1 = 0.2. So our belief here is that P(H) (probability of observing Heads) will be in the interval between 0.4 and 0.6 with probability 0.2.

So between last Wednesday and today we have seen how to express a range of beliefs from knowing nothing to certainty as probability distributions. Starting next Wednesday we will get to the question of how to represent the updating of beliefs when we make new observations.